10.10 矩形棱晶体体积

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  矩形棱晶体体积

  

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  Ben's mom has grown frustrated with Ben's extensive Lego collection thrown all over his bedroom floor.  She gives him a choice of three different boxes in which to store them.  Ben wants to choose the box that will hold the most Legos.  While the boxes are different in dimensions , Ben cannot figure out which will hold the most.  The box measurements are as follows:  
  ::Ben的妈妈对Ben在卧室地板上扔下大量Lego收藏品感到沮丧。 她给了他三个不同的箱子来储存这些物品。 Ben想选择一个能容纳大多数Legos的盒子。 虽然这些箱子的尺寸不同,但Ben无法确定哪个箱子能容纳最多。 方框的测量结果如下:

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  Box A:   height = 5 inches, length = 18 inches, width = 10
  ::方框A:高度=5英寸,长度=18英寸,宽度=10

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  Box B:  height = 8 inches, length = 10 inches, width = 12 inches
  ::方框B:高度 = 8 英寸,长度 = 10 英寸,宽 = 12 英寸

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  Box C:  height = 6 inches, length = 14 inches, width = 10 inches.  
  ::方框C:高度=6英寸,长度=14英寸,宽度=10英寸。

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  Which box has the greatest ?
  ::哪个盒子最棒?

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  In this concept, you will learn how to figure out the volume of rectanglar prisms.
  ::在这个概念中,你将学会如何找出 矩形棱镜的体积。

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  Finding the Volume of a Rectangular Prism
  ::查找矩形棱晶体的量

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  Volume is the measure of how much three-dimensional space an object takes up or holds.
  ::体积是测量一个物体接收或持有的三维空间的量度。

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  Imagine a fish aquarium. Its length, width, and height determine how much water the tank will hold. If you fill it with water, the amount of water is the volume that the tank will hold.  You measure volume in cubic units , because you are multiplying three dimensions: length, width, and height.
  ::想象一下鱼水族馆。 它的长度、 宽度和高度决定着罐体将蓄水多少。 如果您用水填水, 水体的容量就是罐体的容量。 您用立方体体量体积, 因为它乘以三个维度: 长度、 宽度和高度。

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  One way to find the volume of a prism is to consider how many unit cubes it can contain. A unit cube is simply a cube measuring one inch, one centimeter, one foot, or whatever unit of measurement you are using, on all sides. Here are some unit cubes.
  ::找到棱晶体积的一个方法就是考虑它能包含多少个单位立方体。一个单位立方体仅仅是一个立方体,测量一英寸、一厘米、一英尺或任何测量单位。这是一些单位立方体。

  

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  To use unit cubes to calculate volume, simply count the number of unit cubes that fit into the prism. Begin by counting the number of cubes that cover the bottom of the prism, and then count each layer. Let’s see how this works.
  ::使用单位立方体来计算音量, 简单计算适合棱柱的单位立方体数量。 开始计算覆盖棱柱底部的立方体数量, 然后计算每一层。 让我们看看这是怎么回事 。

  

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  How many cubes do you see here? If you count all of the cubes, you will see that there are 24 cubes in this prism.
  ::你在这里看到多少立方体?如果你数了所有的立方体, 你会看到这棱镜里有24个立方体。

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  The volume of this prism is $\begin{array}{r}24{units}^3\end{array}$  or cubic units.
  ::这种棱晶体积为24个单位3或立方单位。

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  Find the volume of the following figure using unit cubes.
  ::使用单位立方体查找下图的音量 。

  

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  How many cubes are in this figure? If you count the cubes, you will get a total of 48 cubes.  The volume of this prism is 48 cubic units or $\begin{array}{r}{\text{units}}^3\end{array}$ .
  ::此图中有多少立方体? 如果计数这些立方体, 你会得到总共48个立方体。 这个棱镜的体积是 48 立方体 或 单位 3 。

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  If you look carefully, you will see that the volume of the rectangular prism is a function of multiplying the length $\begin{array}{r}\times \end{array}$ the width $\begin{array}{r}\times \end{array}$ the height.  You just discovered the formula for finding the volume of a rectangular prism. Now let’s refine that formula a little further. Here is the formula.
  ::如果您仔细看一看, 您就会看到矩形棱柱的体积是乘以长度x宽x高度的函数。 您刚刚发现了查找矩形棱柱体积的公式。 现在让我们对该公式再加一点精细化。 以下是公式 。

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  $\begin{array}{r}V=Bh\end{array}$
  ::V=Bh =Bh

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  The volume is equal to the $\begin{array}{r}B\end{array}$ , base area of the prism, times the height of the prism.
  ::体积等于B,棱晶的基面积,乘以棱晶的高度。

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  Let's look at an example.
  ::让我们举个例子。

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  Find the volume of the prism below.
  ::查找下面棱晶的体积。

  

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  Simply put the values for the length, width, and height in for the appropriate variables in the formula, then solve for $\begin{array}{r}V\end{array}$ , volume.
  ::简单地将公式中相应变量的长度、宽度和高度值放入公式中,然后解析 V, 音量。

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  First find the area of the base. This is the rectangular side on the bottom. Remember, to find the area of a rectangle , multiply the length times the width.
  ::先找到底部的区域。 这是底部的矩形边。 记住, 要找到矩形区域, 请将长度乘以宽度 。

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  $$\begin{array}{rl}B& =lw\\ B& =16\times 9\\ B& =144c{m}^2\end{array}$$

  
  ::B=lwB=16x9B=144平方厘米2

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  The base area is 144 square centimeters. Now multiply this by the height.
  ::基面积为144平方厘米,现在乘以高度。

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  $$\begin{array}{rl}V& =Bh\\ V& =144\times 4\\ V& =576c{m}^3\end{array}$$

  
  ::V=BhV=144x4V=576厘米3

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  You can use the following formula for volume of a rectangular prism.  This combines the two steps that you completed above:
  ::您可以对矩形棱柱的体积使用以下公式。这结合了以上完成的两个步骤:

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  $$\begin{array}{rl}V& =lwh\\ V& =(16)(9)(4)\\ V& =576c{m}^3\end{array}$$

  
  ::V=lwhV=(16)(9)(4)V=576立方厘米

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  The volume of this rectangular prism is 576 cubic centimeters.
  ::这个长方形棱柱的体积是576立方厘米。

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  You can work with the same rectangular prism, but fill it with unit cubes.
  ::您可以用相同的矩形棱柱工作, 但用单位立方体填充它 。

  

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  You can count the unit cubes here to find the volume of the rectangular prism. However, you save time by using the formula for volume.
  ::您可以在此计数单位立方体以找到矩形棱柱的体积。 但是, 您通过使用音量公式来节省时间 。

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  Let's look at another example.
  ::让我们再举一个例子。

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  Find the volume of a container with a length of 15 ft, width of 12 ft, and height of 11 ft.
  ::查找长度为15英尺、宽度为12英尺、高度为11英尺的集装箱的体积。

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  First, plug the values of the dimensions into the formula for volume of a rectangular prism and multiply the values for length and width:
  ::首先,将维度值插入矩形棱柱体积的公式中,并乘以长度和宽度的值:

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  $$\begin{array}{rl}V& =lwh\\ V& =(15)(12)(11)\end{array}$$

  
  ::V=lwhV=( 15) (12) (11)

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  $$\begin{array}{rl}V& =(180)(11)\end{array}$$

  
  ::V=180(11)

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  Next, multiply the results by the value for the height:
  ::下一步,将结果乘以高度值:

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  $$\begin{array}{rl}V& =(180)(11)\\ V& =1,980\end{array}$$

  
  ::V=(180)(11)(11)V=1,980

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  Then, record the answer including the appropriate unit of measurement:
  ::然后记录答案,包括适当的计量单位:

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  $$\begin{array}{rl}V& =1,980{ft}^3\end{array}$$

  
  ::V=1,980平方英尺

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  The answer is the container has a volume of 1,980 cubic feet.
  ::答案是容器的体积为1 980立方英尺。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Ben, who is searching for the box that will hold the most Legos.
  ::早些时候,有人给了你一个问题 关于本, 谁正在寻找的盒子 将持有最多的Legos。

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  Ben needs to figure out which of the following boxes has the greatest volume.
  ::Ben需要弄清楚下面哪个盒子的体积最大

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  Box A:  height = 5 inches, length = 18 inches, width = 10 inches
  ::方框A:高度=5英寸,长度=18英寸,宽度=10英寸

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  Box B:  height = 8 inches, length = 10 inches, width = 12 inches
  ::方框B:高度 = 8 英寸,长度 = 10 英寸,宽 = 12 英寸

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  Box C:  height = 6 inches, length = 14 inches, width = 10 inches
  ::方框C:高度=6英寸,长度=14英寸,宽度=10英寸

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  First, plug the values of the dimensions into the formula for volume of a rectangular prism and multiply the values for length and width:
  ::首先,将维度值插入矩形棱柱体积的公式中,并乘以长度和宽度的值:

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  Box A:    $\begin{array}{rl}V& =(5)(18)(10)\\ V& =(90)(10)\end{array}$
  ::方框A:V=(5)(18)(10)V=(90)(10)

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  Box B:    $\begin{array}{rl}V& =(8)(10)(12)\\ V& =(80)(12)\end{array}$
  ::方框B:V=(8)(10)(12)V=(80)(12)

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  Box C:   $\begin{array}{rl}V& =(6)(14)(10)\\ V& =(84)(10)\end{array}$   
  ::方框C:V=(6)(14)(10)V=(84)(10)

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  Next, multiply the results by the value for the height:
  ::下一步,将结果乘以高度值:

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  Box A:   $\begin{array}{rl}V& =(90)(10)\\ V& =900\end{array}$
  ::方框A:V=(90)(10)(10)V=900

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  Box B:   $\begin{array}{rl}V& =(80)(12)\\ V& =960\end{array}$  
  ::方框B:V=(80)(12)(12)V=960

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  Box C:   $\begin{array}{rl}V& =(84)(10)\\ V& =840\end{array}$   
  ::框C:V=(84)(10)V=840

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  Then, record the answer including the appropriate unit of measurement:
  ::然后记录答案,包括适当的计量单位:

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  Box A:   $\begin{array}{rl}V& =900{in}^2\end{array}$
  ::方框A:V=900英寸2

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  Box B:   $\begin{array}{rl}V& =960{in}^2\end{array}$  
  ::方框B:V=960英寸2

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  Box C:    $\begin{array}{rl}V& =840{in}^2\end{array}$
  ::方框C:V=840英寸2

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  The answer is Box B has the greatest volume and therefore can hold the most Legos.
  ::答案是B箱的容量最大,因此可以容纳最多的Legos。

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  Example 2
  ::例2

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  Carla is cleaning out her fish tank, so she filled the bathtub to the rim with water for her fish to swim in while she empties their tank. If the bathtub is 5.5 feet long, 3.3 feet wide, and 2.2 feet deep, how much water can it hold?
  ::Carla正在清理她的鱼缸,所以她把浴缸填满了水,让鱼在水池里游泳。 如果浴缸长5.5英尺,宽3.3英尺,深2.2英尺,水能支撑多少?

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  First, plug the values of the dimensions into the formula for volume of a rectangular prism and multiply the values for length and width:
  ::首先,将维度值插入矩形棱柱体积的公式中,并乘以长度和宽度的值:

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  $$\begin{array}{rl}V& =lwh\\ V& =(5.5)(3.3)(2.2)\end{array}$$

  
  ::V=lwhV=(5.5)(3.3)(2.2)

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  $$\begin{array}{rl}V& =(18.15)(2.2)\end{array}$$

  
  ::V=(18.15)(2.2)

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  Next, multiply the results by the value for the height:
  ::下一步,将结果乘以高度值:

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  $$\begin{array}{rl}V& =(18.15)(2.2)\\ V& =39.93\end{array}$$

  
  ::V=(18.15)(2.2)V=39.93

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  Then, record the answer including the appropriate unit of measurement:
  ::然后记录答案,包括适当的计量单位:

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  $$\begin{array}{rl}V& =39.93{ft}^3\end{array}$$

  
  ::V=39.93英尺

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  The answer is Carla’s bathtub can hold 39.93 cubic feet of water.
  ::答案是卡拉的浴缸能容纳39.93立方英尺的水。

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  Example 3
  ::例3

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  Find the volume of a container with a length of 10 inches, width of 8 inches, and height of 6 inches.
  ::查找长度为10英寸、宽8英寸、高度为6英寸的容器的体积。

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  First, plug the values of the dimensions into the formula for volume of a rectangular prism and multiply the values for length and width:
  ::首先,将维度值插入矩形棱柱体积的公式中,并乘以长度和宽度的值:

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  $$\begin{array}{rl}V& =lwh\\ V& =(10)(8)(6)\end{array}$$

  
  ::V=lwhV=(10)(10)(8)(6)

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  $$\begin{array}{rl}V& =(80)(6)\end{array}$$

  
  ::V=(80)(6)

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  Next, multiply the results by the value for the height:
  ::下一步,将结果乘以高度值:

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  $$\begin{array}{rl}V& =(80)(6)\\ V& =480\end{array}$$

  
  ::V=(80)(6)(6)V=480

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  Then, record the answer including the appropriate unit of measurement:
  ::然后记录答案,包括适当的计量单位:

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  $$\begin{array}{rl}V& =480{in}^3\end{array}$$

  
  ::V=480英寸3

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  The answer is the container has a volume of 480 cubic inches.
  ::答案是容器的容量为480立方英寸。

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  Example 4
  ::例4

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  Find the volume of a container with a length of 8 meters, width of 7 meters, and height of 3 meters.
  ::找到一个长度8米、宽7米、高度3米的容器的体积。

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  First, plug the values of the dimensions into the formula for volume of a rectangular prism and multiply the values for length and width:
  ::首先,将维度值插入矩形棱柱体积的公式中,并乘以长度和宽度的值:

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  $$\begin{array}{rl}V& =lwh\\ V& =(8)(7)(3)\end{array}$$

  
  ::V=lwhV=(8)(7)(3)

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  $$\begin{array}{rl}V& =(56)(3)\end{array}$$

  
  ::V=(56)(3)

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  Next, multiply the results by the value for the height:
  ::下一步,将结果乘以高度值:

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  $$\begin{array}{rl}V& =(56)(3)\\ V& =168\end{array}$$

  
  ::V=(56)(3)(3)=168

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  Then, record the answer including the appropriate unit of measurement:
  ::然后记录答案,包括适当的计量单位:

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  $$\begin{array}{rl}V& =168m^3\end{array}$$

  
  ::V=168立方米

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  The answer is the container has a volume of 168 cubic meters.
  ::答案是容器的容量是168立方米

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  Review
  ::回顾

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  Find the volume of each rectangular prism. Remember to label your answer in cubic units.
  ::查找每个矩形棱柱的体积。 记住要将答案标签为立方单位 。

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  1. Length = 5 in, width = 3 in, height = 4 in
     ::长度 = 5 英寸,宽 = 3 英寸,身高 = 4 英寸
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  2. Length = 7 m, width = 6 m, height = 5 m
     ::长度=7米,宽度=6米,高度=5米
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  3. Length = 8 cm, width = 4 cm, height = 9 cm
     ::长度 = 8 厘米,宽 = 4 厘米,高度 = 9 厘米
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  4. Length = 8 cm, width = 4 cm, height = 12 cm
     ::长度 = 8 厘米,宽 = 4 厘米,高度 = 12 厘米
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  5. Length = 10 ft, width = 5 ft, height = 6 ft
     ::长度 = 10英尺,宽 = 5英尺,高度 = 6英尺
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  6. Length = 9 m, width = 8 m, height = 11 m
     ::长度=9米,宽度=8米,高度=11米
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  7. Length = 5.5 in, width = 3 in, height = 5 in
     ::长度=5.5英寸,宽度=3英寸,身高=5英寸
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  8. Length = 6.6 cm, width = 5 cm, height = 7 cm
     ::长度=6.6厘米,宽度=5厘米,高度=7厘米
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  9. Length = 7 ft, width = 4 ft, height = 6 ft
     ::长度 = 7 英尺,宽 = 4 英尺,高度 = 6 英尺
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  10. Length = 15 m, width = 8 m, height = 10 m
      ::长度=15米,宽度=8米,高度=10米
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  11. Length = 10.5 m, width = 11 m, height = 4 m
      ::长度=10.5米,宽度=11米,高度=4米
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  12. Length = 12 ft, width = 12 ft, height = 8 ft
      ::长度 = 12 英尺,宽 = 12 英尺,高度 = 8 英尺
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  13. Length = 16 in, width = 8 in, height = 8 in
      ::长度 = 16 英寸,宽 = 8 英寸,身高 = 8 英寸
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  14. Length = 12 m, width = 12 m, height = 12 m
      ::长度=12米,宽度=12米,高度=12米
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  15. Length = 24 in, width = 6 in, height = 6 in
      ::长度 = 24 英寸,宽 = 6 英寸,高度 = 6 英寸

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源